# Questions tagged [dual-spaces]

The dual space of a vector space $V$ over a field $k$ is the vector space of all linear maps from $V$ into $k$.

452 questions
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### Double dual Spaces and the annihilator

I am revising linear algebra, and am a bit stuck on this problem. So, if you define a natural isomorphism $f$ between $V$ and its double dual $V''$, you get $$f:V\rightarrow V''$$ $$v\mapsto E_v$$ ...
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### Adjoint of a linear isomorphic functional is an isomorphism

I have this exercise to solve: Let $E$ and $F$ be normed spaces and let $T \in L(E,F)$, where $L(E,F)$ denotes the set of all bounded linear operators from $E$ to $F$. $T^*$ is the dual operator of ...
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### The dual space of $C(Y,\mathbb R)$ when $Y$ is a complete and separable metric space

Just to be confirmed what is the dual space of $C(Y,\mathbb R)$ i.e the vector space of all continuous functions $f: Y\to \mathbb R$ when $Y$ is complete and separable metric space? Is it the same ...
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### Linear algebra-question about the annihilator for a dual space

I am revising linear algebra and am stuck on a question. I want to show that if $v \notin U$ then there is some $f \in U^0$ such that $f(v) \neq 0$. I am confused about how to go about proving this, ...
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### Hamel dimension of a vector space, and dimension of the dual

I have the following (possibly trivial) observation: Let $K$ be an $\mathbb{F}$-vector space (I believe the argument also works for free modules), and let $X\subseteq K$ be it's basis with ...
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### Describe the Bidual of Real Polynomials

Trying to understand (bi)duals of infinite-dimension vector spaces, I stumbled over the very concrete example of $\mathbb{R}[X]$, the (formal) polynomials over $\mathbb{R}$ or, equivalently, the space ...
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### Connection between a representation and its isomorphic dual

Suppose we have a complex irreducible representation $(V, \phi)$ of a finite group $G$. (Where $\phi : G \rightarrow \text{Aut}_{\mathbb{C}}(V)$). Suppose $V$ it is isomorphic to its dual $V^*$, and ...
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### Representation of the elements of the dual space of the product of topological vector spaces

Assume $(X_i,\mathcal{T}_i)$, $i \in I$ is a family of topological vector spaces and $X:=\prod\limits_{i\in I} X_i$ with the product topology $\mathcal{T}$. Let $\pi_i: X \rightarrow X_i$ be the ...
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